COL863: Quantum Computation and Information Ragesh Jaiswal, CSE, IIT Delhi Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum Computation: Quantum circuits Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum Circuit Controlled operations Theoerm Suppose U is a unitary gate on a single qubit. Then there exist unitary operators A , B , C on a single qubit such that ABC = I and U = e i α AXBXC , where α is some overall phase factor. Question For a single qubit U , can we implement Controlled- U gate using only CNOT and single-qubit gates? Yes Construction sketch The construction follows from the following circuit equivalences. Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum Circuit Controlled operations Question For a single qubit U , can we implement Controlled- U gate using only CNOT and single-qubit gates? Yes Question For a single qubit U , can we implement Controlled- U gate with two control qubits using only CNOT and single-qubit gates? Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum Circuit Controlled operations Question For a single qubit U , can we implement Controlled- U gate using only CNOT and single-qubit gates? Yes Question For a single qubit U , can we implement Controlled- U gate with two control qubits using only CNOT and single-qubit gates? Yes Construction sketch The construction follows from the following circuit equivalence. Here V is such that V 2 = U . Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum Circuit Controlled operations Question For a single qubit U , can we implement Controlled- U gate using only CNOT and single-qubit gates? Yes Question For a single qubit U , can we implement Controlled- U gate with two control qubits using only CNOT and single-qubit gates? Yes Question For a single qubit U , can we implement Controlled- U gate with n control qubits using only CNOT and single-qubit gates? Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum Circuit Controlled operations Question For a single qubit U , can we implement Controlled- U gate using only CNOT and single-qubit gates? Yes Question For a single qubit U , can we implement Controlled- U gate with two control qubits using only CNOT and single-qubit gates? Yes Question For a single qubit U , can we implement Controlled- U gate with n control qubits using only CNOT and single-qubit gates? Yes using ancilla qubits Construction sketch An example construction with n = 4. Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum Circuit Controlled operations A few other gates and circuit identities: Figure: NOT gate applied to the target qubit conditional on the control qubit being 0. Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum Circuit Measurements Principle of deferred measurements Measurements can always be moved from an intermediate stage of a quantum circuit to the end of the circuit; if the measurement results are used at any stage of the circuit, then the clasically controlled operations can be replaced by conditional quantum operations. Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum Circuit Measurements Principle of deferred measurements Measurements can always be moved from an intermediate stage of a quantum circuit to the end of the circuit; if the measurement results are used at any stage of the circuit, then the clasically controlled operations can be replaced by conditional quantum operations. Principle of implicit measurement Without loss of generality, any unterminated quantum wires (qubits which are not measured) at the end of a quantum circuit may be assumed to be measured. Exercise: Suppose ρ is the density matrix describing a two qubit system. Suppose we perform a projective measurement in the computational basis of the second qubit. Let P 0 = I ⊗ | 0 � � 0 | and P 1 = I ⊗ | 1 � � 1 | be the projectors onto the | 0 � and | 1 � states of the second qubit, respectively. Let ρ ′ be the density matrix which would be assigned to the system after the measurement by an observer who did not learn the measurement result. Show that ρ ′ = P 0 ρ P 0 + P 1 ρ P 1 . Also show that the reduced density matrix for the first qubit is not affected by the measurement, that is, tr 2 ( ρ ) = tr 2 ( ρ ′ ). Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum Circuit Measurements Principle of deferred measurements Measurements can always be moved from an intermediate stage of a quantum circuit to the end of the circuit; if the measurement results are used at any stage of the circuit, then the clasically controlled operations can be replaced by conditional quantum operations. Principle of implicit measurement Without loss of generality, any unterminated quantum wires (qubits which are not measured) at the end of a quantum circuit may be assumed to be measured. Exercise: Show that measurement commutes with control. Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum Circuit Universal quantum gates A set of gates is said to be universal for quantum computation if any unitary operation may be approximated to arbitrary accuracy by a quantum circuit involving only those gates. Claim Any unitary operation can be approximated to arbitrary accuracy using Hadamard, phase, CNOT , and π/ 8 gates. Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum Circuit Universal quantum gates Claim Any unitary operation can be approximated to arbitrary accuracy using Hadamard, phase, CNOT , and π/ 8 gates. Proof sketch Claim 1: A single qubit operation may be approximated to arbitrary accuracy using the Hadamard, phase, and π/ 8 gates. Claim 2: An arbitrary unitary operator may be expressed exactly using single qubit and CNOT gates. Claim 2.1: An arbitrary unitary operator may be expressed exactly as a product of unitary operators that each acts non-trivially only on a subspace spanned by two computational basis states (such gates are called two-level gates). Claim 2.2: An arbitrary two-level unitary operator may be expressed exactly using using single qubit and CNOT gates. What about efficiency? Upper-bound: Any unitary can be approximated using exponentially many gates. Lower-bound: There exists a unitary operation that which require exponentially many gates to approximate. Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum Circuit Universal quantum gates Claim 2.1 An arbitrary unitary operator may be expressed exactly as a product of unitary operators that each acts non-trivially only on a subspace spanned by two computational basis states. Proof sketch The main idea can be understood using a 3 × 3 unitary matrix: a d g . U = b e h c f j We will find two-level unitary matrices U 1 , U 2 , U 3 such that U = U † 1 U † 2 U † U 3 U 2 U 1 U = I and 3 Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum Circuit Universal quantum gates Claim 2.1 An arbitrary unitary operator may be expressed exactly as a product of unitary operators that each acts non-trivially only on a subspace spanned by two computational basis states. Proof sketch The main idea can be understood using a 3 × 3 unitary matrix: a d g . U = b e h c f j We will find two-level unitary matrices U 1 , U 2 , U 3 such that U = U † 1 U † 2 U † U 3 U 2 U 1 U = I and 3 Exercise Show that any d × d unitary matrix can be written in terms of d ( d − 1) / 2 two-level matrices. There exists a d × d unitary matrix U which cannot be decomposed as a product of fewer than d − 1 two-level unitary matrices. Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum Circuit Universal quantum gates Claim 2 An arbitrary unitary operator may be expressed exactly using single qubit and CNOT gates. Claim 2.1: An arbitrary unitary operator may be expressed exactly as a product of unitary operators that each acts non-trivially only on a subspace spanned by two computational basis states. Claim 2.2: An arbitrary two-level unitary operator may be expressed exactly using using single qubit and CNOT gates. Proof sketch Let U be a two-level unitary matrix on a n -qubit quantum computer. Let U act non-trivially on the space spanned by the computational basis states | s � and | t � , where s = s 1 , ..., s n and t = t 1 , ..., t n are n -bit binary strings. Let ˜ U be the non-trivial 2 × 2 submatrix of U . Note that we can think ˜ U to be a unitary operator on a single qubit. We will use the gray-code connecting s and t which is a sequence of n -bit strings staring with s and ending with t such that the subsequent strings in the sequence differ only on one bit. Example: s = 101001, t = 110011. g 1 = 101001; g 2 = 101011; g 3 = 100011; g 4 = 110011 Main idea: We will design a sequence of swaps | g 1 � → | g m − 1 � , | g 2 � → | g 1 � , | g 3 � → | g 2 � , ..., | g m − 1 � → | g m − 2 � . We will apply ˜ U to the qubit that differs in g m − 1 and g m . Swap | g m − 1 � with | g m − 2 � , | g m − 2 � with | g m − 3 � and so on. Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
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